Sharp one-third stability of Brenier maps

Proves optimal transport maps from a uniform convex source change at most like the cube root of the target change.

1 paper September 25, 2026 Explained by AI

Optimal transport asks for the cheapest way to move a pile of sand into a new shape. The best plan is a map that tells each grain where to go. If the target shape changes a little, how much can that map change? The paper shows the change is at most a constant times the cube root of the target change. It also shows the cube root cannot be improved, which disproves a conjectured square root.

First step

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Significance

Solid

It finds the exact exponent for a well-studied stability question and disproves a recent conjecture, with a Lean statement of both parts.

Trust

Formal proof listed

Field

Partial differential equations

1 paper

The explanation

Start with sand spread evenly over the square from −1 to 1 in both directions. We send it to three points. Sand with first coordinate below −a goes to (−1, 0). Sand above a goes to (1, 0). The middle strip, of width 2a, goes to (0, 0). Now tilt the strip slightly. Its edges move to |x₁| < a + (a/2)·x₂, and the middle point moves to (0, a/2). The strip keeps the same amount of sand.

How far apart are the two targets? The Wasserstein distance W₂ measures the cheapest way to move one target into the other. Here only the middle point moves, by a/2, carrying mass a. So W₂ = a^{3/2}/2. For a = 0.01 this is 0.0005. How far apart are the two maps? Some sand near the strip edges switches destination by about 1. The average squared gap gives an L² distance of about 0.0500. Shrink a to 0.001. Then W₂ ≈ 0.0000158 and the map gap is ≈ 0.0158.

Divide the map gap by W₂ raised to a power. With power 1/2 the ratio grows: 2.24, then 3.98. With power 1/3 it stays near 0.63. So a square-root bound cannot hold, but a cube-root bound might.

The best map for squared distance is called a Brenier map. A Hölder bound with exponent 1/3 says the map gap is at most C·W₂^{1/3}. Earlier bounds had weaker exponents, such as 2/15 and 1/6, in a related distance W₁Source 1 Source 2. One sharper bound had a constant that depended on the number and masses of the pointsSource 3. Letrouit conjectured a square-root bound for uniform convex sourcesSource 4.

The paper claims the cube-root bound for the uniform measure on any compact convex body, in every dimension d ≥ 2Source 5. The constant does not depend on the number of points, their masses, or their spacingSource 6. The body need not be smoothSource 7. The tilted-strip family above, with three target points, shows no larger exponent works. This disproves the square-root conjectureSource 8 Source 9. The new step is a linear bound, in W₂, on how much the potential function behind the map changesSource 10.

How sure is this? Lean, a computer proof checker, has formal statements of both the cube-root bound and its sharpness. So the paper proves them, as far as the Lean statements match the paper. No specialist review of the paper is published.

What it doesn't show. This does not cover sources that are not uniform on a convex body. It does not give the best constant, and it does not improve bounds stated with the W₁ distance.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

A solid, sharp result with Lean statements of the bound and its sharpness. It covers only uniform sources on convex bodies. No specialist review is published.

Words used

Optimal transport: Finding the cheapest way to move one distribution of mass into another.
Brenier map: The optimal transport map when cost is the squared distance moved.
Wasserstein distance W₂: The square root of the least average squared distance needed to turn one distribution into another.
L² distance: The square root of the average squared gap between two maps.
Hölder bound: A bound of the form C times a distance raised to a fixed power, here 1/3.
Convex body: A bounded closed shape with interior that contains the segment between any two of its points.
Sharp exponent: A power that works, while every larger power fails.

The original paper

Sharp One-Third Stability of Brenier Maps (PDF, opens in a new tab)
September 25, 2026

Lean notes

All quoted lines (10)

M\'erigot, Delalande, and Chazal used dualization to obtain target-uniform bounds from Berman's work, with dimension-dependent exponents, and then proved a dimension-independent $W_1^{2/15}$ bound for a uniform convex source.
Sharp One-Third Stability of Brenier Maps Back to text
Delalande and M\'erigot proved an $L^2$ -map bound of order $W_1^{1/6}$ for source densities bounded above and below on a convex domain and targets in a fixed compact set.
Sharp One-Third Stability of Brenier Maps Back to text
Its constant depends on the number of atoms, a positive lower bound for their masses, and separation and angular geometry, whereas … has one constant for all Borel targets, including degenerating atomic measures.
Sharp One-Third Stability of Brenier Maps Back to text
Letrouit proved an obstruction to exponents above $1/3$ for a nonconvex source and formulated uniform square-root stability in $W_2$ for a uniform convex source as a conjecture.
Sharp One-Third Stability of Brenier Maps Back to text
For the uniform probability measure $\rho$ on a compact convex body in $\R^d$ , $d\ge2$ , quadratic optimal transport maps are one-third H\"older continuous in $L^2(\rho)$ with respect to the target's $2$ -Wasserstein distance.
Sharp One-Third Stability of Brenier Maps Back to text
The constant is independent of atom count, positive atom masses, site separation, and density bounds.
Sharp One-Third Stability of Brenier Maps Back to text
Neither smoothness nor strict convexity of $K$ is required, and $Y$ need not be convex.
Sharp One-Third Stability of Brenier Maps Back to text
A three-atom family on a fixed cube disproves Letrouit's conjectured uniform square-root estimate.
Sharp One-Third Stability of Brenier Maps Back to text
Tilting the middle affine function preserves its cell mass but switches source points between slopes at order-one separation.
Sharp One-Third Stability of Brenier Maps Back to text
Their interpolation inequality is the antecedent of our last step; the improvement here is the linear $W_2$ estimate for potential differences.
Sharp One-Third Stability of Brenier Maps Back to text

Other sources

openai/math, Lean scope note for family 374 The formalization proves one-third Hölder stability of Brenier maps from the uniform measure on a compact convex body with nonempty interior
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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